Abstract: We give new explicit formulas for the representations of the mapping classgroup of a genus one surface with one boundary component which arise fromIntegral TQFT. Our formulas allow one to compute the h-adic expansion of theTQFT-matrix associated to a mapping class in a straightforward way. Truncatingthe h-adic expansion gives an approximation of the representation byrepresentations into finite groups. As a special case, we study the inducedrepresentations over finite fields and identify them up to isomorphism. The keytechnical ingredient of the paper are new bases of the Integral TQFT moduleswhich are orthogonal with respect to the Hopf pairing. We construct theseorthogonal bases in arbitrary genus, and briefly describe some otherapplications of them.